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Discrete Mathematics, Part II
CSE 2353
Fall 2007
Margaret H. DunhamMargaret H. DunhamDepartment of Computer Science and Department of Computer Science and
EngineeringEngineeringSouthern Methodist UniversitySouthern Methodist University
•Some slides provided by Dr. Eric Gossett; Bethel University; St. Paul, Some slides provided by Dr. Eric Gossett; Bethel University; St. Paul, MinnesotaMinnesota•Some slides are companion slides for Some slides are companion slides for Discrete Mathematical Discrete Mathematical Structures: Theory and Applications Structures: Theory and Applications by D.S. Malik and M.K. Senby D.S. Malik and M.K. Sen
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Outline
Introduction Sets Logic & Boolean Algebra Proof TechniquesProof Techniques Counting Principles Combinatorics Relations,Functions Graphs/Trees Boolean Functions, Circuits
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Proof Technique: Learning Objectives
Learn various proof techniques
Direct
Indirect
Contradiction
Induction
Practice writing proofs
CS: Why study proof techniques?
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Proof Techniques Theorem
Statement that can be shown to be true (under certain conditions)
Typically Stated in one of three ways
As Facts
As Implications
As Biimplications
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Validity of Arguments Proof: an argument or a proof of a theorem
consists of a finite sequence of statements ending in a conclusion
Argument: a finite sequence of statements.
The final statement, , is the conclusion, and the statements are the premises of the argument.
An argument is logically valid if the statement formula is a tautology.
AAAAA nn,...,,,,
1321
An
AAAA n 1321...,,,,
AAAAA nn
1321...,,,,
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Proof
A mathematical proof of the statement S is a sequence of logically valid statements that connect axioms, definitions, and other already validated statements into a demonstration of the correctness of S. The rules of logic and the axioms are agreedupon ahead of time. At a minimum, the axioms should be independent and consistent. The amount of detail presented should be appropriate for the intended audience.
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Proof Techniques
Direct Proof or Proof by Direct Method Proof of those theorems that can be expressed in
the form ∀x (P(x) → Q(x)), D is the domain of discourse
Select a particular, but arbitrarily chosen, member a of the domain D
Show that the statement P(a) → Q(a) is true. (Assume that P(a) is true
Show that Q(a) is true By the rule of Choose Method (Universal
Generalization), ∀x (P(x) → Q(x)) is true
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Proof Techniques Indirect Proof
The implication P → Q is equivalent to the implication ( Q → P)
Therefore, in order to show that P → Q is true, one can also show that the implication ( Q → P) is true
To show that ( Q → P) is true, assume that the negation of Q is true and prove that the negation of P is true
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Proof Techniques Proof by Contradiction
Assume that the conclusion is not true and then arrive at a contradiction
Example: Prove that there are infinitely many prime numbers
Proof: Assume there are not infinitely many prime numbers,
therefore they are listable, i.e. p1,p2,…,pn
Consider the number q = p1p2…pn+1. q is not divisible by any of the listed primes
Therefore, q is a prime. However, it was not listed. Contradiction! Therefore, there are infinitely many
primes.
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Proof Techniques Proof of Biimplications
To prove a theorem of the form ∀x (P(x) ↔ Q(x )), where D is the domain of the discourse, consider an arbitrary but fixed element a from D. For this a, prove that the biimplication P(a) ↔ Q(a) is true
The biimplication P ↔ Q is equivalent to (P → Q) ∧ (Q → P)
Prove that the implications P → Q and Q → P are true Assume that P is true and show that Q is true Assume that Q is true and show that P is true
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Proof Techniques
Proof of Equivalent Statements Consider the theorem that says that statements
P,Q and r are equivalent
Show that P → Q, Q → R and R → P Assume P and prove Q. Then assume Q and
prove R Finally, assume R and prove P
What other methods are possible?
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Other Proof Techniques
Vacuous
Trivial
Contrapositive
Counter Example
Divide into Cases
Constructive
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Proof Basics
You can not prove by example
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Proof Strategies with Quantifiers
Existential Constructive
some mathematicians only accept constructive proofs Nonconstructive
show that denying existence leads to a contradiction
Universal to prove false:
one counter-example to prove true:
usually harder the choose method
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Proofs in Computer Science
Proof of program correctness
Proofs are used to verify approaches
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Mathematical Induction Assume that when a domino is knocked over, the next domino is knocked over by it Show that if the first domino is knocked over, then all the dominoes will be knocked
over
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Mathematical Induction
Let P(n) denote the statement that then nth domino is knocked over
Base Step: Show that P(1) is true Inductive Hypothesis: Assume some P(i) is
true, i.e. the ith domino is knocked over for some
Inductive Step: Prove that P(i+1) is true, i.e.1i
)1()( iPiP
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Outline
Introduction Sets Logic & Boolean Algebra Proof Techniques Counting PrinciplesCounting Principles Combinatorics Relations,Functions Graphs/Trees Boolean Functions, Circuits
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Learning Objectives
Learn the basic counting principles—multiplication and addition
Explore the pigeonhole principle
Learn about permutations
Learn about combinations
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Basic Counting Principles
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Basic Counting Principles
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Pigeonhole Principle
The pigeonhole principle is also known as the Dirichlet drawer principle, or the shoebox principle.
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Pigeonhole Principle
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Permutations
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Permutations
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Combinations
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Combinations
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Generalized Permutations and Combinations